High removal rate magnetorheological finishing head

ABSTRACT

A magnetorheological finishing head comprising magnetic pole pieces, nozzle shape, and wheel shape tailored to maximize volumetric removal rate. The carrier wheel for a ribbon of magnetorheological fluid is aspherical, preferably a toroid having a short radius perpendicular to, and the long radius parallel to, the axis of rotation, although the shape of the wheel may be any aspherical or free form parallel to the wheel&#39;s axis of rotation, e.g., toroidal or cylindrical. A magnetic field is generated by shaping the pole pieces to create a substantially uniform magnetic field over a defined gap therebetween such that the field strength in the area of the fluid ribbon is uniform. The nozzle has a non-circular opening to provide a fluid stream having a width that covers the width range of the magnetic field. It is the combination of these three features that allows for a novel MRF removal function.

FIELD OF THE APPLICATION

The present invention is directed to systems for magnetorheologicalfinishing of substrate surfaces; more particularly, to amagnetorheological finishing head comprising a rotatable work surface(the outer surface of an equatorial section of a “wheel”) disposedbetween opposing magnetic pole pieces and capable of carryingmagnetorheological fluid (MR fluid) on the surface of the wheel into andthrough a work zone (the “spot”) between the wheel and the work surfaceof the substrate (“workpiece”) being finished, wherein the MR fluid is“stiffened” by being subjected to a magnetic field exerted by themagnetic pole pieces which may be electromagnets or permanent magnets,and material is removed from the substrate surface through abrasion bythe stiffened MR fluid; and most particularly, to such amagnetorheological finishing head wherein the rotatable work surface isnon-spherical, the magnetic pole pieces generate a substantially uniformmagnetic field, and the MR fluid is presented to the work zone on therotatable work surface as a wide ribbon of fluid material.

BACKGROUND OF THE INVENTION

Use of magnetically-stiffened magnetorheological fluids for abrasivefinishing and polishing of substrates is well known. Such fluids,containing magnetically-soft abrasive particles dispersed in a liquidcarrier, exhibit magnetically-induced plastic behavior in the presenceof a magnetic field. The apparent viscosity of the MR fluid can bemagnetically increased by many orders of magnitude, such that theconsistency of the MR fluid changes from being nearly watery to being avery stiff abrasive paste. When such an abrasive paste is directedappropriately against a substrate surface to be shaped or polished,e.g., an optical element, a very high level of finishing quality,accuracy, and control can be achieved.

U.S. Pat. No. 5,795,212, “Deterministic magnetorheological finishing”,issued Aug. 18, 1998 to Jacobs et al., discloses a method and apparatusfor finishing a workpiece surface using MR fluid wherein the workpieceis positioned near a carrier surface such that a converging gap isdefined between a portion of the workpiece surface and the carriersurface. A magnetic field is applied substantially at the gap, and aflow of stiffened MR fluid is introduced into the gap such that a workzone is created in the MR fluid, thereby forming a sub-aperturetransient finishing tool for engaging and causing material removal atthe portion of the workpiece surface. The workpiece or the work zone ismoved relative to the other to expose different portions of theworkpiece surface to the work zone for predetermined time periods toselectively finish portions of the workpiece surface to predetermineddegrees.

U.S. Pat. No. 5,839,944, “Apparatus for deterministic finishing ofworkpieces”, issued Nov. 24, 1998 to Jacobs et al., discloses a methodand apparatus for finishing a workpiece surface using MR fluid whereinthe workpiece is positioned near a carrier surface such that aconverging gap is defined between a portion of the workpiece surface andthe carrier surface. A magnetic field is applied substantially at thegap, and a flow of stiffened MR fluid is introduced into the converginggap such that a work zone is created in the MR fluid, thereby forming asub-aperture transient finishing tool for engaging and causing materialremoval at the portion of the workpiece surface. The workpiece or thework zone is moved relative to the other to expose different portions ofthe workpiece surface to the work zone for predetermined time periods toselectively finish portions of the workpiece surface to predetermineddegrees.

U.S. Pat. No. 5,951,369, “System for magnetorheological finishing ofsubstrates”, issued Sep. 14, 1999 to Kordonski et al., discloses animproved system for increasing the effectiveness of magnetorheologicalfinishing of a substrate. An inline flowmeter is close-loop linked tothe rotational speed of a pressurizing pump to assure that the flow ofmagnetorheological fluid to the work zone is constant. A simplifiedcapillary viscometer is disposed in the fluid delivery system at theexit thereof onto the wheel surface. Output signals from the flowmeterand the viscometer pressure sensor are sent to a computer whichcalculates the viscosity of MRF being delivered to the work zone andcauses replenishment of carrier fluid to the work-concentrated MR fluidto return the viscosity to aim to assure that a constant concentrationof magnetic solids is being provided to the work zone. Asymmetric polepieces for the field magnet at the work zone extend the magnetic fieldalong the wheel surface upstream of the work zone to permit fullmagnetic stiffening of the MRF before it engages the work piece, whileminimizing fringing field in the vicinity of the viscometer, and toshorten the magnetic field along the wheel surface downstream of thework zone.

U.S. Pat. No. 6,106,380, “Deterministic magnetorheological finishing”,issued Aug. 22, 2000 to Jacobs et al., discloses method and apparatusfor finishing a workpiece surface using MR fluid wherein the workpieceis positioned near a carrier surface such that a converging gap isdefined between a portion of the workpiece surface and the carriersurface. A magnetic field is applied substantially at the gap, and aflow of stiffened MR fluid is introduced into the converging gap suchthat a work zone is created in the MR fluid, thereby forming asub-aperture transient finishing tool for engaging and causing materialremoval at the portion of the workpiece surface. The workpiece or thework zone is moved relative to the other to expose different portions ofthe workpiece surface to the work zone for predetermined time periods toselectively finish the portions of the workpiece surface topredetermined degrees.

U.S. Pat. No. 6,506,102, “System for magnetorheological finishing ofsubstrates”, issued Jan. 14, 2003 to Kordonski et al., discloses animproved system for magnetorheological finishing of a substratecomprising a vertically oriented bowl-shaped carrier wheel having ahorizontal axis. The carrier wheel is preferably an equatorial sectionof a sphere, such that the carrier surface is spherical. The wheelincludes a radial circular plate connected to rotary drive means andsupporting the spherical surface which extends laterally from the plate.An electromagnet having planar north and south pole pieces is disposedwithin the wheel, within the envelope of the sphere, and preferablywithin the envelope of the spherical section defined by the wheel. Themagnets extend over a central wheel angle of about 120 degrees, suchthat magnetorheological fluid is maintained in a partially stiffenedstate ahead of and beyond the work zone. A magnetic scraper removes theMR fluid from the wheel as the stiffening is relaxed and returns it to aconventional fluid delivery system for conditioning and re-extrusiononto the wheel. The system is useful in finishing large concavesubstrates, which must extend beyond the edges of the wheel, as well asfor finishing very large substrates in a work zone at the bottom deadcenter position of the wheel.

U.S. Pat. No. 8,944,883, “System for magnetorheological finishing of asubstrate”, issued Feb. 3, 2015 to Kordonski, discloses a sphericalwheel meant for carrying a magnetorheological finishing fluid andhousing a variable-field permanent magnet system having north and southiron pole pieces separated by primary and secondary gaps with acylindrical cavity bored through the center. A cylindrical permanentmagnet magnetized normal to the cylinder axis is rotatably disposed inthe cavity. An actuator allows rotation of the permanent magnet to anyangle, which rotation changes the distribution of flux in the magneticcircuit through the pole pieces. Thus, one can control field intensityin the gaps by positioning the permanent magnet at whatever angleprovides the required field strength. Because the field also passesabove the pole pieces, defining a fringing field outside the wheelsurface, the variable field extends through a layer of MR fluid on thewheel, thus varying the stiffness of the MR fluid as may be desired forfinishing control.

In all of these prior art references, the disclosed wheel is anequatorial spherical section; a method to tailor the shape of themagnetic field by shaping the tips of the pole pieces is not disclosed;and method and apparatus for shaping the cross-sectional area of themagnetorheological fluid ribbon on the wheel with a non-circular nozzleexit is not disclosed.

Prior art magnetorheological finishing heads have a limit in materialremoval rate that is driven by two primary factors. The first factor isthe fluid being used in the process, which drives the peak removal rate.The second factor is the apparatus that makes up the finishing head,which drives both the peak and the volumetric removal rate.

The geometry of the apparatus limits the physical size of the removaltool and in many cases limits the deterministic magnetorheologicalfinishing process. In particular, prior art processes that focus onfinishing large optics and/or inducing an aspheric shape to a sphericalsurface can take hours or even days of finishing to reach the desiredfinal figure. In such cases, having a larger removal function providesthe opportunity to significantly reduce the cycle time of the polishingrun.

In the prior art, larger removal functions have been created simply byincreasing the diameter of the spherical magnetorheological finishingwheel, but in many cases a larger wheel is not possible or practical andalso is very costly to fabricate with the required precision. As thewheel gets larger, the same precision is still required for wheelrunout, and achieving the desired tolerances becomes significantlyharder and more expensive.

Considering the factors controlling removal rate, the work zone must beenlarged both in the direction of wheel travel and transverse to thedirection of wheel travel to increase the removal rate. It is known thatthe spot can be lengthened by increasing the wheel radius, and that thepeak removal rate can be increased by making the spot deeper between thewheel and the substrate workpiece. What is not known in the prior art ishow to increase the volumetric removal rate by making the spot wider,preferably without increasing substantially the radius or other geometryof the wheel.

What is needed in the art is a magnetorheological finishing head havinga tailored magnetic field, nozzle shape, and wheel shape that maximizesthe volumetric removal rate of substrate material.

SUMMARY OF THE INVENTION

A magnetorheological finishing head comprises magnetic pole pieceshaving specifically shaped opposing tips; a non-circular nozzle shape;and a non-spherical wheel shape and surface to maximize volumetricremoval rate. The carrier wheel for a ribbon of MR fluid isnon-spherical and is preferably an equatorial section of a toroid havinga short radius about an axis parallel to, and a long radius about anaxis perpendicular to, the axis of rotation of the wheel, although theshape of the wheel may be any aspherical or free form shape having anaxis of rotation parallel to the wheel's axis of rotation, e.g.,toroidal or cylindrical. A tailored magnetic field is generated byshaping the tips of the pole pieces to create a magnetic fringing fieldover a defined gap therebetween greater than a prior art gap such thatthe field strength is maintained at a useful intensity over the width ofthe ribbon. The magnets may be either electromagnets or permanentmagnets, although typically electromagnets are employed as in the priorart. The nozzle has a non-circular opening to provide a fluid stream ina width that covers the range of the magnetic field. It is thecombination of these three features that allows for a maximum increasein MRF removal function, although these features taken singly or inpairs can provide a significant increase in MRF removal function overthat of the prior art.

The present invention creates an opportunity for material removal ratesat least four times greater than do prior art systems.

The system is especially useful for conducting low order figurecorrections on large substrates and introducing shape change to anoptical surface such as aspheric generation.

Further features and advantages of the present invention will becomeapparent to those of ordinary skill in the art in view of the drawingsand detailed description of preferred embodiments below.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will now be described, by way of example, withreference to the accompanying drawings, in which:

FIG. 1 is an elevational cross-sectional view of a portion of a priorart magnetorheological finishing head;

FIG. 2a is an elevational cross-sectional view of a portion of amagnetorheological finishing head in accordance with the presentinvention;

FIG. 2b is an elevational view of the portion of a magnetorheologicalfinishing head shown in FIG. 2a showing in addition an MR fluid ribbonon the wheel surface, a workpiece in position for material removal, andwork zone or “spot” therebetween;

FIG. 3 is a perspective view from above of the magnetorheologicalfinishing head shown in FIGS. 2a and 2 b;

FIG. 4 is an elevational view of a first embodiment of a nozzle inaccordance with the present invention;

FIG. 5 is a cross-sectional pan view of the nozzle shown in FIG. 4;

FIG. 6 is an elevational cross-sectional view of the elements of amagnetorheological work zone;

FIG. 7 is an elevational cross-sectional view of a portion of an MRfinishing head showing dimensions of a ribbon of MR fluid in accordancewith the present invention;

FIG. 8a is a diagram showing the relationship of a toroid formed inaccordance with the present invention to a finishing wheel taken as anequatorial section of the toroid;

FIG. 8b is a diagram like that shown in FIG. 8a showing orthogonallyintersecting arcs on the surface of a toroid formed in accordance withthe present invention, the arcs having respective radii R₁ and R₂,wherein R₁≠R₂;

FIG. 9 is an isometric view of a first embodiment of magnetic polepieces in accordance with the present invention;

FIG. 10 is an isometric view of a second and preferred embodiment ofmagnetic pole pieces in accordance with the present invention;

FIG. 11 is a cross-sectional view of the magnetic field resulting fromsimply moving prior art parallel pole planes apart in an effort to widenthe magnetic field and hence the width of the work zone;

FIG. 12 is a cross-sectional view of the magnetic field resulting fromforming the opposing pole surfaces as conic sections in an effort towiden the magnetic field and hence the width of the work zone;

FIG. 13 is a cross-sectional view of the magnetic field resulting fromforming the opposing pole surfaces as toroidal sections in an effort towiden the magnetic field and hence the width of the work zone;

FIG. 14 is a graph showing idealized magnetic field lines from FIGS.11-13;

FIG. 15 is a plan view of material removal rate in a typical prior artwork zone; and

FIG. 16 is a plan view of typical material removal rate in a work zoneproduced by a magnetorheological finishing head formed in accordancewith the present invention.

DETAILED DESCRIPTION OF THE INVENTION

Referring to FIGS. 1 and 6, a portion of a prior art magnetorheologicalfinishing head 10 comprises a finishing wheel 12 having a disc-shapedcentral portion 14 supporting an equatorial spherical finishing portion16 having a finishing surface 18. Wheel 12 is mounted for rotation onaxle 20 carried in precision bearings 22 a,22 b. Axle 20 is driven aboutaxis of rotation 30 by an electric motor system (not shown). Below andadjacent to finishing portion 16 and on opposite sides of disc-shapedcentral portion 14 are first and second magnet pole pieces 24 a,24 b,preferably identical but of opposite polarities, i.e., north and south.These pole pieces typically have planar opposing faces 26 a,26 b set ata predetermined first spacing from each other. Polepieces 24 a,24 b maybe electromagnets or permanent magnets.

When the electromagnets are energized, a magnetic fringing field (notshown) is formed through and above finishing portion 16 wherein a ribbonof MR fluid 17 being carried on surface 18 is stiffened to a pasteconsistency. A substrate 21 to be finished, e.g., a lens as shown inFIG. 6, is positioned, typically for rotation about its own axis 23,above the wheel surface at a distance from the wheel less than thethickness of the incoming MR fluid ribbon, thus creating a converginggap and forming a work zone or “spot” 19 wherein abrasive finishing ofsubstrate 21 disposed in work zone 19 is carried out. The dimensions ofthe converging gap may be varied according to the requirements of aspecific finishing application. In FIG. 6, the height of the ribbonentering the work zone is RH, the plunge depth of the work piece intothe ribbon is D, and the resulting gap G between the work piece and thewheel surface is the thickness of the work zone 19.

Referring now to FIGS. 2a, 2b , 6,8, and 9-13, an improvedmagnetorheological finishing head 110 for forming a wider and longerwork zone is substantially identical with the prior artmagnetorheological finishing head 10 shown in FIG. 1 except that theupper corners of the magnetic pole pieces 124 a,124 b are modified asshown. Preferably, the upper corners are rounded 128 a,128 b as shown inFIGS. 2a , 10, and 13 or beveled 126 a,126 b and 226 a,226 b as shown inFIGS. 9 and 12 and may be of any desired shape, e.g., conical, curvedwith a radius, or freeform. The actual values of radius and spacingbetween pole pieces 124 a,124 b may be selected as required to form aspecific size work zone for any particular application. It has beenfound that providing a rounded or beveled shape can result in a fringingfield 40,240 having lateral uniformity over a substantially greaterwidth than that formed by the prior art pole piece arrangement shown inFIG. 1. Preferably, the curved shapes 128 a,128 b are formed as portionsof a torus in accordance with the equations shown hereinbelow regardingthe shape of the finishing wheel surface, in particular a ring toruswherein the distance from the center of the tube to the center of thetorus is larger than the radius of the tube.

Referring still to improved magnetorheological finishing head 110, asdescribed hereinabove, finishing portion 116 having finishing surface118 is formed as a non-sphere, preferably a toroid having a short radius119 perpendicular to, and a long radius coincident with, the axis ofrotation 130, although the shape of the wheel may be any aspherical orfree form parallel to the wheel's axis of rotation 130, e.g., toroidalor cylindrical (toroid with infinite long radius). An advantage of thisgeometry is that it allows for larger removal functions without asignificant increase in the size of the overall tool, i.e., the diameterof the wheel. Another advantage is that the toroidal wheel allows theremoval function to get wider without requiring an increase in thevolume of the fluid that a prior art spherical wheel requires. Thisfeature helps reduce the need for higher flow rates and larger pumpingsystems to achieve an equivalent result.

Referring to FIGS. 2a, 8a, and 8b , a finishing wheel 116 having surface118 may be more generally defined as a surface of revolution other thanspherical. FIGS. 8a,8b show an idealized shape 142 having a first radiusR₁ being rotated about a second radius R₂ to form a three-dimensionalshape 144 wherein R₁ and R₂ create respective orthogally-intersectingarcs A₁ and A₂ on wheel surface 118. (Note that when R₁=R₂, the wheelsurface is spherical as in the prior art.) In simplest form, shape 142is a circle and shape 144 is a torus, but it is possible that ahigher-order polynomial or other equation can be used to define asurface that can be revolved around R₂. For higher removal rates R₁should be much larger than R₂. These values can be chosen based on twofactors: 1) the shape of the optic to avoid the geometry of the wheelinterfering with the geometry of the workpiece (concave optics inparticular), and 2) the larger the radius R₂ the wider (and thus larger)the removal function will be for a given MRF flowrate.

In explicit form, the wheel geometry may be expressed by:

Z=f(x,y)=R_(y)±√[R_(y)−g(x))²−y²], where g(x) is the generating curveand Z is the algebraic shape of the wheel.

For a torus:

g(x)=R_(x){1−√[1−(x/R_(x) ²]}, where g(x) is a circle with radius R_(x).

Referring to FIGS. 3 through 5, the present invention requires a shapechange to the MRF ribbon formed on finishing surface 118. Prior art MRFribbons are created using a round nozzle exit of a specified innerdiameter. A typical ribbon shape is round when extruded from a prior artexit port having a diameter of 3 mm and a cross-sectional area of 7.3mm².

To increase the size of the removal function (work zone) the need is toincrease its width. A wider removal function requires an MRF stream thatis spread out laterally and injected on the wheel across the area thatcovers the width of the removal function before the MRF ribbon 150reaches the work zone, typically at the top-dead-center position of thewheel. Thus, if the nozzle exit is non-circular, and preferably isshaped as a slot, the MR fluid is spread out prior to landing on thewheel, allowing for wider removal functions.

Nozzle assembly 132 comprises a feed tube 134 entered into a housingblock 136 and terminating in a distributor 138 within housing block 136that discharges into an internal slot formed at the desired width of theMRF ribbon to be generated and terminating at an exit slot 140. In apresently preferred embodiment, exit slot 140 is about 19 mm wide andabout 0.9 mm high, resulting in an aspect ratio greater than 20. Thecross-sectional area of this design is 17.8 mm², allowing nearly two anda half times the flow rate of the prior art nozzle when operated at thesame delivery pressure. Increased flow is required to generate a widerremoval function by filling a larger area between the wheel andsubstrate. Preferably, the ends of the slot are rounded to avoidstagnant zones in the corners and unwanted buildup of fluid.

Obviously, other slot shapes and dimensions may be selected as may berequired for specific finishing applications, e.g., the “slot” may beformed by a line of discharge holes rather than a continuous slot, orthe slot may be non-uniform in height.

The height and width of the ribbon may be manipulated on the wheel afterextrusion. The angle of incidence of the fluid jet onto the wheel caninfluence the ribbon width: as the nozzle extrusion angle increases fromtangential toward perpendicular, the ribbon tends to spread laterally onthe wheel. Increasing the wheel velocity to beyond the “flow matching”value at which the fluid jet velocity matches the wheel's tangentialvelocity causes the fluid to be stretched out, resulting in a lowercross-sectional area of the ribbon. The benefit of spreading the ribbonout allows the operator to manage the overall height of the ribbon andthe dimensions shown in FIG. 6 to achieve a wide removal function. Oncethe fluid ribbon is energized by the magnetic field, the abrasiveboundary layer 19 (work zone) is generated across the width of theribbon.

Preferably, the height of a ribbon of magnetorheological fluid on afinishing wheel when entering a work zone is between 1.20 mm and 1.56mm, the plunge depth into said ribbon of magnetorheological fluid by aworkpiece being finished by the magnetorheological finishing head isbetween 0.60 mm and 0.81 mm, and a gap between the work piece and thefinishing wheel is between 0.60 mm and 0.75 mm.

FIGS. 3 and 7 show a ribbon 150 of width W and thickness RH disposed onwheel surface 118.

Referring now to FIG. 11, it is seen that simply moving the prior artplanar-facing pole pieces 26 a,26 b farther apart than the standardspacing shown in FIG. 1 creates a magnetic field 140 in the work zonethat is laterally non-uniform and somewhat weaker in the center,resulting in an undesirable bimodal removal function. Alternatively(FIGS. 9, 12 and 14), beveling the pole pieces as with conical faces 226a,226 b results in a fairly uniform field 240 overall with a slightlylower field intensity. Referring now to FIGS. 10,13 and 14 with a radiuson the pole pieces 124 a,124 b results in a fairly uniform field 40overall with a higher field intensity.

Referring now to FIG. 14, idealized magnetic fields just above the wheelsurface are shown for the conditions disclosed hereinabove in FIGS.11-13.

Referring now to FIGS. 15 and 16, a prior art work zone spot 55 (FIG.15) is shown in comparison to a work zone spot 155 achievable by amagnetorheological finishing apparatus in accordance with the inventionas shown in FIG. 2a . A typical prior art spot 55, from a spherical 150mm diameter wheel, has a width 60 of about 4.0 mm and a length 70 ofabout 10.0 mm, thus having a working area of about 40.0 mm², whereas animproved spot 155 may have a width 160 of about 18.0 mm and a length 170of about 21.0 mm, thus having a working area of about 378.0 mm²,providing a removal rate many times larger than a prior art spot.

Thus, the present invention comprises three novel elements: a) magnetpole pieces having rounded upper corners, b) a non-spherical wheelfinishing surface, preferably toroidal, and c) an MRF application nozzlehaving a non-circular exit. It is the combination of these threefeatures that allows for a maximum increase in MRF removal function,although these features taken singly or in pairs can provide asignificant increase in MRF removal function over that of the prior art.

Various changes may be made to the structure and method embodying theprinciples of the invention. The foregoing embodiments are set forth inan illustrative and not in a limiting sense. The scope of the inventionis defined by the claims.

1. A magnetorheological finishing head, comprising, a) a rotatablefinishing wheel having a non-spherical finishing surface; b) first andsecond magnetic pole pieces of opposing polarity having corners disposedwithin said finishing wheel and having opposing faces, wherein thecorners of said opposing faces closest to said finishing surface have ashape selected from the group consisting of conical, beveled, toroidal,radial, and freeform; and, c) a nozzle assembly terminating in anon-circular exit.
 2. A magnetorheological finishing head, comprisingany one of the following three elements: a) a rotatable finishing wheelhaving a non-spherical finishing surface; b) first and second magneticpole pieces of opposing polarity disposed within said finishing wheeland having opposing faces, wherein the corners of said opposing facesclosest to said finishing surface have a shape selected from the groupconsisting of conical, beveled, toroidal, radial, and freeform; and c) anozzle assembly terminating in a non-circular exit.
 3. Amagnetorheological finishing head, comprising any two of the followingthree elements: a) a rotatable finishing wheel having a non-sphericalfinishing surface; b) first and second magnetic pole pieces of opposingpolarity having corners disposed within said finishing wheel and havingopposing faces, wherein the corners of said opposing faces closest tosaid finishing surface have a shape selected from the group consistingof conical, beveled, toroidal, radial, and freeform; and, c) a nozzleassembly terminating in a non-circular exit.
 4. The magnetorheologicalfinishing head in accordance with claim 1 wherein the shape of saidnon-spherical finishing surface is selected from the group consisting oftoroidal, cylindrical, and free-form.
 5. The magnetorheologicalfinishing head in accordance with claim 1 wherein said magnetic polepieces are components of a magnetic system selected from the groupconsisting of electromagnet and permanent magnet.
 6. Themagnetorheological finishing head in accordance with claim 1 wherein amagnetic field above said finishing surface is substantially uniformfrom edge to edge of said magnetic field.
 7. The magnetorheologicalfinishing head in with claim 1 wherein said non-circular exit is a slot.8. The magnetorheological finishing head in accordance with claim 1wherein said rotatable finishing wheel is formed in accordance with theformulaZ=f(x,y)=R _(y)±√[(R _(y) −g(x))²-y ²], where g(x) is the generatingcurve and Z is the algebraic definition of said rotatable finishingwheel.
 9. The magnetorheological finishing head in accordance with claim1 wherein said first and second magnetic pole pieces are formed suchthat when they are energized a uniform magnetic fringing field is formedover a desired width on said rotatable finishing wheel.
 10. Themagnetorheological finishing head in accordance with claim 1 whereinsaid nozzle assembly is formed such that a ribbon of magnetorheologicalfluid extruded therefrom is of uniform thickness from edge to edge ofsaid ribbon.
 11. The magnetorheological finishing head in accordancewith claim 1 wherein said non-circular exit of said nozzle assembly isselected from the group consisting of a slot, a slot with rounded ends,and a plurality of holes.
 12. The magnetorheological finishing head inaccordance with claim 2 wherein the shape of said non-sphericalfinishing surface is selected from the group consisting of toroidal,cylindrical, and free-form.
 13. The magnetorheological finishing head inaccordance with claim 2 wherein said magnetic pole pieces are componentsof a magnetic system selected from the group consisting of electromagnetand permanent magnet.
 14. The magnetorheological finishing head inaccordance with claim 2 wherein a magnetic field formed above saidfinishing surface is substantially uniform from edge to edge of saidmagnetic field.
 15. The magnetorheological finishing head in accordancewith claim 2 wherein said first and second magnetic pole pieces areformed such that when they are energized a uniform magnetic fringingfield is formed over a desired width on said rotatable finishing wheel.16. The magnetorheological finishing head in accordance with claim 2wherein said nozzle assembly is formed such that a ribbon ofmagnetorheological fluid extruded therefrom is of uniform thickness fromedge to edge of said ribbon.
 17. The magnetorheological finishing headin accordance with claim 3 wherein the shape of said non-sphericalfinishing surface is selected from the group consisting of toroidal,cylindrical, and free-form.
 18. The magnetorheological finishing head inaccordance with claim 3 wherein said magnetic pole pieces are componentsof a magnetic system selected from the group consisting of electromagnetand permanent magnet.
 19. The magnetorheological finishing head inaccordance with claim 3 wherein a magnetic field formed above saidfinishing surface is substantially uniform from edge to edge of saidmagnetic field.
 20. The magnetorheological finishing head in accordancewith claim 3 wherein said first and second magnetic pole pieces areformed such that when they are energized a uniform magnetic fringingfield is formed over a desired width on said rotatable finishing wheel.21. The magnetorheological finishing head in accordance with claim 3wherein said nozzle assembly is formed such that a ribbon ofmagnetorheological fluid extruded therefrom is of uniform thickness fromedge to edge of said ribbon.